An Elementary Treatise on Plane and Solid GeometryJ. Munroe and Company, 1847 - 150 σελίδες |
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Σελίδα xvi
... Proof . If the fractions of a proportion A : B = C : D are reduced to a common denominator , they give AX D BX C BX D BX D or , omitting the common denominator , AX DBX C. This proposition is called the test of proportions . Theorem II ...
... Proof . If the fractions of a proportion A : B = C : D are reduced to a common denominator , they give AX D BX C BX D BX D or , omitting the common denominator , AX DBX C. This proposition is called the test of proportions . Theorem II ...
Σελίδα xvii
... Proof . The application of the test to the preceding proportion gives the B2AX D , square root of which is B = L ( A × D ) . A succession of several equal ratios is called a continued proportion . Thus , A : B = C : D = E : F = & c . is ...
... Proof . The application of the test to the preceding proportion gives the B2AX D , square root of which is B = L ( A × D ) . A succession of several equal ratios is called a continued proportion . Thus , A : B = C : D = E : F = & c . is ...
Σελίδα xviii
... Proof . The proportion gives , by the preceding proposition , A : B = C : D = = A - C : B - D A + C : A - C = B + D : B D. A + C : B + D whence , by transposing the means , Theorem VI . The sum of the first two terms of a pro- portion ...
... Proof . The proportion gives , by the preceding proposition , A : B = C : D = = A - C : B - D A + C : A - C = B + D : B D. A + C : B + D whence , by transposing the means , Theorem VI . The sum of the first two terms of a pro- portion ...
Σελίδα 6
... Proof . Thus , if P and M ( fig . 2 ) are in the same di- rection from A , the two straight lines AP and AM must likewise , by § 15 , have the same direction , and must con- sequently coincide in the same straight line . 18. Axiom . A ...
... Proof . Thus , if P and M ( fig . 2 ) are in the same di- rection from A , the two straight lines AP and AM must likewise , by § 15 , have the same direction , and must con- sequently coincide in the same straight line . 18. Axiom . A ...
Σελίδα 8
... Proof . For it is equal to the sum of the two right angles APM , MPE , formed by the perpendicular PM . 26. Theorem . The sum of all the successive an- gles APB , BPC , CPD , DPE , and EPA ( fig . 10 ) , formed in a plane about a point ...
... Proof . For it is equal to the sum of the two right angles APM , MPE , formed by the perpendicular PM . 26. Theorem . The sum of all the successive an- gles APB , BPC , CPD , DPE , and EPA ( fig . 10 ) , formed in a plane about a point ...
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Άλλες εκδόσεις - Προβολή όλων
Συχνά εμφανιζόμενοι όροι και φράσεις
ABC fig adjacent angles angle BAC arc BC base and altitude bisect centre chord circumference common altitude construct convex surface Corollary DEF fig Definitions denote diameter divided Draw equal arcs equal distances equiangular with respect equilateral equivalent frustum given angle given circle given line given polygon given sides given square gles greater half the product Hence homologous sides hypothenuse infinite number infinitely small Inscribed Angle inscribed circle isosceles Let ABCD line AB fig line BC lines drawn mean proportional number of sides oblique lines parallel lines parallel to BC parallelogram parallelopipeds perimeter perpendicular plane MN polyedron polygon ABCD &c Problem Proof pyramid or cone radii radius rectangles regular polygon right triangle Scholium sector segment side BC similar polygons similar triangles solid angle Solution sphere spherical polygon spherical triangle straight line tangent Theorem triangles ABC triangular prism vertex vertices whence
Δημοφιλή αποσπάσματα
Σελίδα 68 - The perimeters of two regular polygons of the same number of sides, are to each other as their homologous sides, and their areas are to each other as the squares of those sides (Prop.
Σελίδα 127 - Every section of a sphere, made by a plane, is a circle, Let AMB be a section, made by a plane, in the sphere whose centre is C.
Σελίδα 71 - Rectangles of the same altitude are to each other as their bases, and rectangles of the same base are to each other as their altitudes. 245.
Σελίδα 20 - The sum of the three angles of any triangle is equal to two right angles.
Σελίδα xv - The first term of a ratio is called the antecedent, and the second term the consequent.
Σελίδα 83 - ... we suppose the error A to be of any magnitude whatever. 286. Definition. Similar sectors and similar segments are such as correspond to similar arcs. 287. Theorem. Similar sectors are to each other as the squares of their radii. Proof. The similar sectors AOB, A'OB ' (fig. 136) are, by the same reasoning as in t5 97, the same parts of their respective circles, which the angle O= O...
Σελίδα 31 - Theorem. In the same circle, or in equal circles, equal arcs are subtended by equal chords.
Σελίδα 87 - To construct a parallelogram equivalent to a given square, and having the sum of its base and altitude equal to a given line.
Σελίδα 99 - B, from the plane. 320. Theorem. Oblique lines drawn from a point to a plane at equal distances from the perpendicular are equal; and of two oblique lines unequally distant the more remote is the greater.
Σελίδα 78 - Similar triangles are to each other as the squares of their homologous sides. Proof. In the similar .triangles ABC, A'B'C